Never rely on a simulation for real ice safety. Ice strength depends on temperature history, snow, currents, springs, cracks, age and local conditions that no model captures. Treat every number here as a teaching estimate; the real rules of thumb come from local authorities.
Live numbers
1. Ice growth (Stefan's law)
Heat conducted up through the ice freezes more water at the bottom. With the surface at air temperature and the bottom at 0 °C:
rho_i L dh/dt = dT / ( h/k_i + R_s )
h = sqrt( 2 k_i dT t / (rho_i L) ) (bare ice)
h^2/(2k_i) + R_s h = dT t/(rho_i L) (with snow)
k_i = 2.2 W/mK, rho_i = 917 kg/m3, L = 334 kJ/kg (typical values). Snow of depth h_s adds a series resistance R_s = h_s/k_s + 1/h_c, with k_s = 0.15 W/mK (dry snow, typical 0.1-0.3) and an air film h_c = 15 W/m2K (illustrative). 8 cm of snow slows growth several times.
2. Black ice vs white ice
Clear black ice freezes from the water; white ice (snow-ice, slush) is porous and about half as strong. Snow load can flood the ice when h_s rho_s > h (rho_w - rho_i); the white fraction rule used here, 0.05 + 0.35 F, is an illustrative simplification. Effective thickness: h_eff = h_black + sqrt(0.5) h_white.
3. Load bearing (floating elastic plate)
Gold: P [kg] = A h_eff^2 [cm] A = 3.5 safe, up to 7 short-term
w = P / (8 sqrt(k D)) k = rho_w g, D = E h^3 / (12 (1 - nu^2))
l = (D / k)^(1/4) characteristic length of the dip
P_safe uses A = 3.5 (conservative); the break-through estimate uses A = 7. Safety factor SF = P_safe / load. An auger hole reduces capacity by 1 - 0.3 min(1, d/4h) and thaw weakening multiplies by 1 - 0.35 min(1, melt/10 cm): both are illustrative rules, not measured laws. Cracks fade in from 40 % of the break-through load and are fully shown at 70 %. E = 5 GPa (typical 3-6).
4. Verdicts
OK: SF ≥ 2. CAUTION: 1 ≤ SF < 2. UNSAFE: SF < 1. BREAKS: load above the break-through estimate. NO ICE: under 5 cm.
Honest simplifications
Constant air temperature over the frost days; no wind, sun, currents, springs or water flow; uniform thickness; a static load (moving and repeated loads, vehicles near shore and dynamic resonance are worse); the ice top is kept at a fixed height instead of floating freeboard.
Independent numerical checks
Stefan closed form vs RK4 time stepping, Gold's formula vs the textbook table, and a Westergaard-type point-load stress estimate against the A = 3.5-7 band.
Credits
Hut, angler and sled: generated with Meshy, cleaned and re-baked in Blender 5.2. Pond, ice, cracks, trees: Blender 5.2 with Cycles-baked textures. Three.js r160 + WebGL. Not a safety tool.